With air superiority, your army may operate as it sees fit, having freedom of the battlefield. Without it, you will likely lose battle after battle as your troops are not only bombarded from above, but their movements are anticipated and countered.
Example of the latter: The Ottoman Empire's attempt to take the Suez Canal in 1915 via overland march. The movements of their approaching army were reported to the British & Indian troops by the new "wonder weapon" of the age - the aeroplane. The Turks lost 2000 men, the British 150.
> If we turn our attention to point 3 [What is going to prevent the RN from interfering?] for a while, the standard response is to say that the Luftwaffe could sink the RN ships. However, the Luftwaffe of the period had a pathetic record against warships. 39 RN destroyers took part in the Dunkirk evacuation. This operation required manoeuvring in a small harbour, with periods stationary while embarking troops. The Luftwaffe had command of the air for long periods. In these ideal conditions, the Luftwaffe managed to put out of commission a grand total of 4 destroyers. 4 out of 39 does not bode well for the Luftwaffe's chances.
Addressed pretty solidly I think, everyone except the German air force did well against navies. Air superiority doesn't matter if they can't destroy the targets on the ground/sea.
At no point in the war was the Luftwaffe capable of (a) clearing the English Channel for such an invasion and (b) keeping the channel clear for reinforcements. It might have been possible to get an initial landing party on the shores of England, but reinforcing them would have been extraordinarily difficult.
The Luftwaffe would have been harassed not only by surface ships with rather formidable anti-aircraft systems, but also by the RAF which they continuously failed to knock out of the war.
Let's also not forget that Germany possessed no vessels truly capable of taking an invasion force across the Channel to begin with.
For the Pacific theater, sure, but at the time the Luftwaffe were notoriously bad at attacking ships, especially the hundreds of fast-moving smaller gunboats which would have wrecked the invasion barges.
I have read that after his graduation from West Point in 1944, Lt. John Eisenhower got to visit with his father. On one of the beaches, watching the dense motor and foot traffic, Lt. Eisenhower said, "You couldn't do this if you didn't have air supremacy." Gen. Eisenhower snorted and said "I wouldn't be here if I didn't have air supremacy."
